Finally, the formulas of transformation from this last system of rectangular again to polar co-ordinates, the origin being the same, the fixed plane the plane of x'y' and the polar axis the axis of x', are Now, if the values of x, y, z, x', y', z', given by (3) and (5) be substituted in (4), we have at once the following system of equations: cos a = cos e cos b+ sin c sin b cos a sin a cos B sin c cos b cos c sin b cos A sin a sin B = (6) which are the known fundamental formulas of spherical trigonometry, but established without imposing any restrictions upon the values of the parts of the triangle. From this investigation it appears that these formulas may be regarded as formulas of transformation from one system of polar co-ordinates to another, or rather from one system of spherical co-ordinates to another. For example, the co-ordinates of a star referred to the pole of the equator and the meridian of a place whose colatitude is c, are its polar distance a, and its hour angle в; the co-ordinates of the same star referred to the pole of the horizon and the meridian, are its zenith distance b, and its azimuth ; and the formulas (6) express the relations by means of which we can pass from one of these systems to the other. values of the sides and angles (5.) Let us now inquire what are the corresponding in the series of triangles expressed by (1) and (2). values of the parts of one of these triangles, which, if we please, we may suppose to be the triangle whose parts are less than . Then since sin (2n+4)=sin p, cos (2 n = + 4) = cos ¢, the equations (6) will be satisfied by the substitution of 2 n ≈+a, 2 n n + b, &c., for a, b, and c; and therefore the triangle (a, b, c, A, B, C) is the first of an infinite series obtained from it by the successive addition of 2 to each or all of its parts, every triangle of the series being such, that the relations of its parts are expressed by (6), when a, b, c, A, B, C, are assumed to represent those parts. It is evident, also, from the principle of "uniformity of direction" observed in the preceding demonstration in reckoning the sides and angles, that we must be able to satisfy the equations, by making either all the sides, or all the angles, or all the sides and angles, negative at the same time,* and, considering each of the triangles * The student will do well to conceive the position of the angular points of the triangles on the surface of the sphere with these variations. N. B. That the sides are all negative together, or the angles together, or both together. The same thing stated in the text may be made evident by referring to equations thus obtained as the first of a series, as above, we have three more series. We have then the four series following: In all the terms of these series, n may have the same or different values; and we thus have all the possible combinations of the values represented by (1) and (2), so long as m in (2) is even. But if we substitute 2 n + 1 for m we shall find that the following series will satisfy the equations (6) the 6th, 7th, and 8th of which series are derived from the 5th, as the 2d, 3d, and 4th were derived from the 1st, in the preceding paragraph. By successively exchanging a for b and c, we find eight more series, namely, (6), which involve all the relations of the six elements of a spherical triangle, and which will be satisfied by changing simultaneously a, b, and c into a,b,c, or A, B, C, into A,— B, — c or both; observing the general rule that sin (—) =— sin and cos (p) = cos p. *The student will try these elements given in the 5th series, in eqs. (6), observing that cos {(2 n + 1) = + 9 } = cos (180° + $) =—cos ₫ and sin {(2 n + 1) x + ♦ } = - sin . 5. Since three great circles by their mutual intersections (provided they have not a common diameter), divide the surface of the whole sphere into eight primitive triangles (whose parts are all less than ), the three angular points of each of which give sixteen triangles, whose parts are all less than 2,* therefore, three great circles of the sphere form in general one hundred and twenty-eight triangles, each of which may be considered as the first term of an infinite series of triangles formed from it by the successive addition of 2 to each or all of its parts. AMBIGUITY IN THE SOLUTION OF THE GENERAL SPHERICAL TRIANGLE. For the sake of brevity, I shall call the spherical triangle, whose parts are only limited by the condition 360°, the general spherical triangle. Although any three points of the surface of the sphere may be regarded (in general) as the angular points of sixteen such triangles, yet to the problem" given three parts of the triangle to find the other three," there will in every case be but two solutions, i. e. two triangles containing the same data. From the equations (6), and the consequences that flow from them, we can always obtain expressions for both the sine and cosine of each of the required parts, which would fully determine the triangle, were it not that in every case one of these expressions at least involves a radical of the second degree, and has either two different numerical values, or two values numerically equal with opposite signs. To avoid this ambiguity it was thought expedient to limit all the parts of the triangles to values less than 1800, or to consider only the simple geometrical triangle. By this means all the cases in which the required quantities can be found by a cosine or tangent, without involving radicals, become fully determined. But this occurs in but four of the six cases, the other two still having two solutions; so that although six conditions were thus imposed, three limiting the data themselves, and three the quæsita, the object of removing all ambiguity was not reached. * i. e., making n = 0, in each of the 16 series of the last art. We shall see from the solutions of the general triangle, that the ambiguity is entirely removed in every case by the imposition of a single condition restricting the sign of either the sine or cosine of but one of the required parts. The general method here, as in many parts of the mathematics, is therefore the simplest. FORMULAS REQUIRED FOR THE SOLUTION OF THE GENERAL SPHERICAL TRIANGLE. 1. As the formulas (6) are the same as those deduced in trigonometrical works for limited spherical triangles, we may avail ourselves, for the solution of the general triangle, of all the formulas (found in those works), deduced from them in a general manner. It is not necessary therefore to repeat all those deductions here; but I shall add a demonstration of GAUSS's equations, slightly differing from the common one, in order to establish them in their generality. then the products p q, pr, p s, qr, q s, rs, are respectively equal to the products PQ, PR, PS, Q R, Q S, Rs. To demonstrate this, we have only to form the following equations, which are easily deduced from the fundamental formulas :— sin c (cos à COS B) = (1 cos c) sin (a + b)† (1 cos c) sin (A + B) sin c (cos b± cos a)‡ which, transformed by the formulas of the trigonometric analysis,§ give respectively sin c cos Acos a sin b―sin a cos b cos c sin c cos B = cos b sin a-sin b cos a sin c By addition and subtraction of these we obtain sin c (cos Acos B) = (sin a cos b sin b cos a) (1 = cos c) whence the formula in the text. This is obtained from the last by substituting -c &c., for c to produce the supplemental or polar triangle. To wit: formulas which express the factors of the above forms in terms of the sines and cosines ofc, c, † (A ± B), } (a+b) by means of which we have the following, each of the above forms furnishing two: 2 sinc cos c2 sin † (A + B) cos } (A — B) = 2 sinc cos † c 2 sin ↓ (a + b) cos (a - b) 2 siac cosc 2 sin (A-B) COS } (A + B) =2 sinc cosc 2 sin (a - b) cos (a+b) &c., &c. 3. The same notation being employed, the quantities p2, q2, r2, s2, are respectively equal to the quantities r2, Q, R2, s2. For we have p q X p r = P Q X P R and q r =Q R, whence by division p2 = p2, and in the same way q2= q2, r2 = R2 s2 = s2. 4. GAUSS'S EQUATIONS. From the preceding paragraph we deduce In these equations the positive sign must be taken in all the second members, or the negative sign in all of them. For if we take p = +P, the equations p q = P Q, p r = P R, p s = P s, being divided by this; give q= +Q, r=+R, 8s. But if we take p—P, the same equations divided by this, give qq, r = — R, s ——S. Hence the two following groups of equations, the first group comprising those commonly known as GAUSS's equations, which are identical with (I.), (II.), (III.) and (IV.) of Art. 86, by clearing the latter of fractions. 5. Now when the parts of the triangle are limited to values less than 1800, the second of these groups is excluded, since cosc, sin (A+B), cosc, cos (a — b) are then all positive. But when the triangle is unlimited, both groups must be admitted, and the question arises, when are we to employ the positive, and when the negative sign? GAUSs himself has remarked (Theoria Mot. Corp. Cal., Art. 54), that cases occur in practice in which it is necessary to employ the negative sign, and promises elsewhere a fuller explanation, which, however, I have not been able to find. But the nature of these cases and the answer to the question above propounded will be easily inferred from the following considerations. We have seen that the formulas (6) apply not only to the triangle whose parts a, b, c, A, B, C, are all less than 360°, or 2, but also to all the triangles whose parts are 2 n + a, 2 n x + b, 2 n = + c, 2 n π + s, 2 n я + в 2n + c, n being π The first two of the six that would result are all that we have thought necessary to write. They are identical with |